A genus-2 crossing equation in $d\geq 2$

TL;DR

The paper develops a genus-2 crossing framework for CFTs in $d\ge 2$, linking two complementary channel decompositions (sunrise and dumbbell) of the genus-2 partition function on $M_2=(S^1\times S^{d-1})\sharp (S^1\times S^{d-1})$ and deriving corresponding Casimir equations for the blocks. By matching the two channels and analyzing the thermal flat limit, it isolates a one-dimensional data function $h(z)$ encoding squares of thermal one-point coefficients and derives an explicit heavy-heavy-heavier OPE asymptotic relation in 3d, controlled by Casimir-based saddles and shadow formalism. The work also clarifies the role of Weyl factors and the mapping class group in 3d, and lays the groundwork for a genus-2 numerical bootstrap and future explorations of Lorentzian/low-lying-saddle physics. Overall, the genus-2 crossing equation provides a new, highly constrained arena to extract and relate CFT data across dimensions, with direct ties to thermal physics and potential computational approaches. The methodology combines geometric channel decompositions, Casimir-guided saddle analysis, and shadow-based inversion to connect OPE data with thermal coefficients, offering a path toward richer non-perturbative constraints on CFT spectra and OPE structures.

Abstract

We explore a "genus-2" crossing equation obeyed by CFTs in general dimensions $d\geq 2$. This crossing equation relates two different decompositions of the "genus-2 partition function" -- namely the partition function on the connected sum $M_2=(S^1\times S^{d-1})\sharp (S^1\times S^{d-1})$. The "sunrise" channel decomposition expresses $M_2$ as a pair of three-punctured spheres glued together with cylinders, while the "dumbbell" channel decomposition expresses $M_2$ as a gluing of two one-point functions on $S^1\times S^{d-1}$. We introduce coordinates to describe each channel, and write down Casimir equations obeyed by the corresponding blocks. We also explain why equality between the two channels guarantees mapping class group invariance of the genus-2 partition function in 3d CFTs. As an application of the genus-2 crossing equation, we derive a novel relation between asymptotics of "heavy-heavy-heavier" OPE coefficients and squares of thermal one-point coefficients in 3d CFTs. Along the way, we demonstrate how expectation values of conformal generators can help locate saddle points in large quantum number limits.

Figures (7)

• Figure 1: Left: the genus-2 crossing equation obtained by starting with the four-point crossing equation (solid lines), and contracting the external operators (indicated by dashed lines). The left-hand side of the resulting equation is the "sunrise" channel (since the diagram looks like a sunrise Feynman diagram), and the right-hand side is the "dumbbell" channel. Right: the resulting crossing equation in terms of geometry, drawn in 2d for simplicity. A resolution of unity is inserted at each gray circle, giving a sum over genus-2 blocks in the sunrise channel and a sum over squared torus one-point functions in the dumbbell channel.
• Figure 2: Cartoon illustrating the thermal flat limit of the genus-2 manifold with colored circles and lines representing different $d-$dimensional balls. In the upper graph, the two strips in the right diagram represent two copies of $S^1\times \mathbb{R}^{d-1}$ where the lines in the front and back are identified as suggested by the arrows. The lower graph represents the dumbbell channel geometry. In the thermal flat limit, the two smaller balls are tangential to the larger one but they are not tangent to each other.
• Figure 3: In the dumbbell channel, we obtain $M_2$ by removing two unit balls $B_L$ and $B_R$ from two pieces of $S^1\times S^{d-1}$ and glue their boundaries with a cylinder of inverse temperature $\beta$. The two blue lines represent two non-contractable loops with length $2r\sinh(\beta_L/2r)$ and $2r\sinh(\beta_R/2r)$ . In $d>2$, the two red loops can be shrunk towards poles of $S^{d-1}$.
• Figure 4: The correspondence of loops and conformal group elements in different channels. In the sunrise channel (right) the $a$-loop (red) is mapped to the group element $g_1^{-1} g_3$, the loop $b$ (blue) is mapped to $g_3^{-1} g_2$ and the loop $a\cdot b$ (green) is mapped to $g_1^{-1} g_2$. In the dumbbell channel (left),the loop $a$ is mapped to $g_M^{-1} g_L^{-1} g_M$, the loop $b$ is mapped to $g_R$ and the loop $a\cdot b$ is mapped to $g_M^{-1} g_L^{-1} g_M g_R$.
• Figure 5: Matching of loops between the two channels. In the sunrise channel geometry (right picture), the red, blue and green circles have length $\beta_{13},\beta_{23}$ and $\beta_{12}$. The corresponding loops in the dumbbell channel geometry are drawn in the left picture. The two rectangular strips represent two pieces of $S^1\times \mathbb{R}^{d-1}$ with the vertical directions being the $S^1$ direction. The upper edges of the two strips are all identified with the lower edges, as indicated by the arrow. By carefully tracing the identifications, one can check that the green lines in the left picture form a closed loop. The height of the cylinder in the right picture becomes infinitesimally small in the thermal flat limit, but with a generic value of $z$, the two horizontal segments of the green loop will always have a finite length. The blue, red circle will shrink to size zero and two hot spots will develop at $P_{13},P_{23}$. In the dumbbell channel, the two hot spots get blown up to two pieces of flat space.